Proposal to revise a hypothetical learning trajectory on estimation

for inviting more students to use estimation, we should take into account the following factors:

- Mathematical problem themselves

This includes: kind of problems, operation of numbers, and type of numbers. Kind of problems is meant as a classification of problems: whether the problems are complete (all information is given), incomplete, or unavailable data.

Based on the research results of the second research period, we found that estimation problems with incomplete or unavailable data are generally more difficult than problems with complete data (see section 5.2). This result is in line with the proposal from Van den Heuvel-Panhuizen (2001)—that is estimation problems with incomplete or unavailable data are generally more difficult than problems with complete data. Thus, for the purpose of learning-teaching we should give problems with complete data first.

Operation of numbers means mathematical operations which are used for solving problems: addition, subtraction, multiplication, or division.

Mathematically seen, for the same type, magnitudes of numbers, and context, estimation problems in addition or subtraction are easier than in multiplication or division. Thus, for the purpose of learning-teaching we should give problems with addition or subtraction first.

Type of numbers can mean: integer numbers, rational numbers, real numbers, etc. In general, problems which include only integers are relatively easier than problems with fractions or decimal numbers (rational numbers) or even combinations of these. Consequently, for learning-teaching purposes, we should give problems which include integer numbers first, then problems with other types or combinations of these.

- Design of problems

This includes: difficulties, context of problems, openness of problems, selection of numbers, and type of questions.

Difficulties of problems mean the number of solution steps of problems. A problem which needs more than one step solution is generally more difficult than another problem which only needs one step. In our research, we can take an example: Problem 9.a and 9.b (see Table 4.1) are difficult for students because these need more than one step solution. Thus, for a revision of the HLT 2, we should take into account at number of problem step solutions.

Regarding context of problems, we think that to get students to use estimation, we should design problems with contextual situations that are experientially real for students, fit with the students’ age, or fit with the students’

world. Therefore, the problems can be perceived by them. For example, in our research, the ice cream context is best fit for students because they are used to facing such problems in their daily life. On the other hand, supermarket context (like a receipt context) is less experientially real for students—especially in Indonesia. Because supermarket problems are usually faced by adults: especially in Indonesia, the supermarket context more suitable to mothers. Thus, for a revision of our designed problems, we might need to look for other contexts that

are really experientially real for students to replace several problems that are not really fit.

To revise the HLT, we should also consider about the openness of the problems. In our example, the ice cream problem is an open problem with different answers and strategies. Therefore, the problem can invite students to use more different estimation strategies. Thus, we need to change several problems, from closed problems to open problems—with careful considerations, for example, either from designer problems experts or from experienced teachers.

The selection of numbers means that numbers are used in the problems should be close to nearest tens, hundreds, thousands, or other easy numbers because such numbers are able to invite more students to use estimation. For example, we can see Problems 5 – 8 (see Table 5.2), where these problems used numbers near to tens, hundreds, thousands. In practice, for example, rather than using a number like 1,675 it is better to use 1,950 to catch students to rounding off the number to 2,000. Thus, in the case of our problems, we need to revise several numbers in the Problems 1 – 4 (see Table 5.2) to numbers that are close to nearest tens, hundreds, or thousands.

According to Van den-Heuvel Panhuizen (2001), three types of questions that can be used to invite students to use estimation strategies are: Are there enough? (2) Could this be correct? (3) Approximately how much is it? In the case of our problems, in this research, we used only the first and the second type of questions. Thus, to revise the HLT 2, we need to add problems with the third type of questions.

- Classroom cultures

This includes: students’ own productions and strategies, teacher guidance, group and classroom discussion (interactivity in learning-teaching situations).

Referring to the tenets of RME, students should be guided in explorations and solving problems of contextual problems within an interactive classroom. In our case, we tried to implement the tenets of RME and its principles; however it was not really attained. In the case of Problems 9 – 12, in our view the teacher has

given too much intervention, namely giving examples how to solve the problems by an exact calculation and giving clues how to find solutions of inkblot and unavailable data problems.

In the case of Problem 5 that is relatively successful in inviting more students to use estimation, we see from the video recordings and field notes analysis (can be found in section 5.2), the teachers’ role and the lesson structure of classroom learning-teaching situation are very important.

Thus, to revise the HLT, we should prepare the teacher better to make sure whether the teacher understands in giving guidance to his/her students or not. In case the teacher wants to give help, she/he should understand entirely the topic that will be taught during the learning-teaching processes. Thus, a good teacher’s preparation before lessons is important. Besides that, classroom learning-teaching structure in the form of group and class discussion might motivate students to share: opinion, justification, and strategies. Therefore, a good classroom management is very important to be considered by teachers.

Therefore, the HLT should include not only instructional materials but also teachers’ preparation explicitly (on understanding the topics, problems, materials, and classroom management).